Theorem f1orn 6600

Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1orn	⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		dff1o2 6595	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹))
2		eqid 2798	. . 3 ⊢ ran 𝐹 = ran 𝐹
3		df-3an 1086	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹))
4	2, 3	mpbiran2 709	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 4	bitri 278	1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ^◡ccnv 5518  ran crn 5520  Fun wfun 6318   Fn wfn 6319  –1-1-onto→wf1o 6323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331

This theorem is referenced by:  f1f1orn  6601  infdifsn  9104  efopnlem2  25248  cycpmcl  30808